Palindromes On The 12 hour Clock
By: Robleh Wais
9/3/15
It is common knowledge that the 12-hour cycle of a clock leads to palindromic configurations predictably. I am becoming fascinated by how these palindromes occur and would like to explore the attributes of them. With this in mind let us look at the structure of palindromes that occur as a 12-hour clock proceeds through its cycles.
I will define two terms to divide the palindromes on the 12-hour clock set:
A full palindrome is one in which the numbers are all the same, like 1:11, 2:22, 4:44.
A half palindrome is one where the numbers form a palindrome by an alternating sequence of x and y variables, where x and y are integers from 0 to 9. Examples would be xyx, for instance, 1:21, 3:13, or 4:24. If we don't allow 0 to be considered an initial placeholder, there are only 2 four digit half palindromes, e.g. 10:01 and 12:21 for the 12-hour clock.
There is an ongoing challenge to discover if any pair of integers when reversed and added will eventually generate a palindrome. If we take integers larger than 2 digits many quickly become palindromes, while a few like 196 don't. 196 is one fundamental example of a number resisting converging to a palindrome. A term called a Lychrel number was coined by Wade VanLandingham. VanLandingham is famous for calculating to 300 million digits, the integer 196 without the occurrence of palindrome. Visit his site here for a deeper look at this unique integer. http://www.p196.org/
a Lychrel number is one, that does not converge to a palindrome after extremely long iterations of reversed addition. The problem with 196, and other integers that occur like 196, is there is no deductive methodology to show that a number like 196 will become a palindrome. That is, there is no constructive proof of the conjecture that some integers never form palindromes if they are reversed and added repetitively. A mathematician would call this a decision procedure. Happily, in this pursuit, I'm not trying to construct a proof of palindromic occurrence due to the operation of addition on reverse integers. What I want to investigate is if we can derive palindromic sets from a 12-hour clock, what happens if we
take the output and continue the process? This kind of feedback process occurs in abstract algebra all the time. Look at the dichotomy of palindromic types specified above. All the possible palindromic sequences for the 12-hour clock can be cataloged as an initial set. Since time-keeping on the 12-hour clock is structured such that we won't use zero (0) as a placeholder, many palindromic sequences that could be used will not, like for instance 03:30, 01:10, 02:20, etc. Still, there are many possible palindromic combinations. Moreover, since we are restricting our domain to a limited set of numbers, due to the nature of addition for the 12-hour clock, we
have a very neat and predictable collection of palindromic numbers. But, not if we expand the derived sets to include two forms of addition. From this point on, I will abandon consideration of palindromic types as specified above, not relevant at present, but maybe later, since this is a living document.
Look at any given hour on the 12-hour clock. For example, 1 o'clock to 2 o'clock. If we tabulate it, we'll find that there is a definite set of palindromes in this interval. If we do this for every hour on the 12-hour clock we have the following tabulation:
57 minutes are palindromic for the hours from 1 AM to the following 1 AM
The above interval is a two-hour period, and within this interval, the following palindromes occur:
2 minutes have a palindrome for the hours 10 to 12.
1 minute has a palindrome for the hour 12 am to 1 am.
54 minutes are palindromic from 1 am to 9 pm, which comes to 6 per hour (9 hours).
We can see that for a twelve-hour period, 57 minutes are palindromic. What is becoming much more interesting now, is the surjective sets that form, if we take the palindromic output of each hour starting with 1 am to the succeeding 1 am and under the operation of addition, combine them with the previous palindromic number. This would be taking the Fibonacci numbers of the 12-hour clock times. In parallel to this, we can add the palindromic numbers that occur hour-to-hour. Two different output sets of palindromes will be derived and may well have some cyclic properties. These sets are surjective mappings because the original set is being mapped to elements within it to create a new non-isomorphic set. It is a mapping ONTO a new set that the operation of addition (+) creates from the original set. Some algebraic sets are surjective. The natural and base 10 logarithms are surjective. So is modular arithmetic. It is important to note that, algebraically this is a surjective map and not an injective or bijective map. In the latter case, we are relating sets by different interpretations of set mapping. A bijective function simply forms an isomorphic map between elements of two sets, like 2↔B. Where the two elements are in one-to-one correspondence to each other. An injective map would map UNIQUE elements of one set to another via isomorphism. Please note a bijective function maps the same elements of a set to each other more than once without violating the notion of a function. They just have to be the same elements mapped. 9 Mod 3 =0 is a bijective map that maps 3 to itself thrice both ways. However, a surjective map does violate the notion of a function in that it can map the SAME element of one set to differing elements of the output set. And this is what we should expect on the 12-hour clock. 3:13 AM is identical to 3:13 PM numerically, so its image in a new set related by addition and would be a mapping to the same elements as on the original clock.
I've begun an Excel worksheet to do these operations. The worksheet so far has derived 14 additive palindromes and 11 Fibonacci palindromes. This is exciting because there is no way to tell where this operation will go. This sheet was simple to construct and it was manually tabulated. As mentioned above, we have a small well-ordered set (not in the ordinal set theory sense of that phrase) so I didn't need to write macros to get results. If I go through a 2nd iteration of the derived palindromes are there be more? What if I mix the palindromic types by addition? What if I take the derived palindromes and recombine them with the originals? The permutations are many as interested readers might have guessed. This is an unfinished endeavor, and I plan to go further with the Excel sheet in the future. My intuition is iterative operations on these sets might lead to infinite palindromic reoccurrence.
You can download this file by clicking the icon below. Update: this file is now a macro-embedded file that has calculated palindromes on the 12-clock to the 3rd iteration, and counting...
Excel file for 12-Hour Clock Palindromes Palindromes on the 12 Hour Clock
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It is important to note that the above Excel file illustrates just how random the derived palindromes are. The occurrence of a palindrome is not generated by a definable mapping of object sets.
Thus, any proof that they will occur as a result of a binary operation, on a ring of integers generated by the 12-hour clock, is not possible by the usual methods of set theory.